70. Number is a collection of units; it is a general idea, because to conceive the number, we do not need to know of what class, or how many the units may be. The idea of number in general abstracts absolutely all such determinations. It is evident that, whatever number we imagine, we can always conceive another still greater, and if we assign a limit to a number, we can always remove it indefinitely, so that the limit of one is not the limit of the other. To the idea of number, we unite the idea of a limit and of the negation of another limit. Therefore, if we unite to the idea of number in general, the idea of the negation of limit in general, we shall obtain the idea of an infinite number. 71. What does this idea represent? It represents nothing determinate: it is an entirely abstract conception, formed of two other abstract conceptions, those of number and the negation of limit. No determinate object corresponds to it; it is a work of our understanding referred to objects in general, without a determination of any sort. We may now solve the difficulties previously intimated. 72. Why is a series of terms presented to us as infinite, which, when we examine it closely, we find wants some of the marks of infinity? Because, in the first instance, we apply the negation of limit under a condition which we take no notice of in the second instance. Set us the series _a_, _b_, _c_, _d_, _e_, .......... It is evident that we may continue it infinitely, and conceive the negation of all limit of this continuation: in this sense, the number of terms is infinite; for the idea of the negation of limit is really applied to the series. When we ask if the number of terms is absolutely infinite, we abstract the condition under which we had united the negation of limit. That, therefore, which is infinite in one instance is not so in another. Still there is not any contradiction because the yes and the no refer to different suppositions. 73. Let us take a line and measure it by feet. Producing this line we multiply the number of feet; and we may conceive the negation of all limit of this multiplication. The number of feet will then be infinite. If instead of a foot we take an inch as the unit of measure, we shall have a number twelve times as great. This number would also be infinite, and thus we should have two infinite numbers, one of them greater than the other. Is there any contradiction in this? Certainly not: there is only a different combination of ideas. In the first case, the idea of the negation of limit was subordinated to the condition of the division of the line into feet: whereas, in the second case, we introduce a different condition; the division of the line into inches.